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<h1>Minimize thermal noise power of an array with arbitrary 2-D geometry</h1>
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<span class="comment">% "Convex optimization examples" lecture notes (EE364) by S. Boyd</span>
<span class="comment">% "Antenna array pattern synthesis via convex optimization"</span>
<span class="comment">% by H. Lebret and S. Boyd</span>
<span class="comment">% (figures are generated)</span>
<span class="comment">%</span>
<span class="comment">% Designs an antenna array such that:</span>
<span class="comment">% - it has unit a sensitivity at some target direction</span>
<span class="comment">% - obeys constraint for minimum sidelobe level outside the beamwidth</span>
<span class="comment">% - minimizes thermal noise power in y (sigma*||w||_2^2)</span>
<span class="comment">%</span>
<span class="comment">% This is a convex problem described as:</span>
<span class="comment">%</span>
<span class="comment">%   minimize   norm(w)</span>
<span class="comment">%       s.t.   y(theta_tar) = 1</span>
<span class="comment">%              |y(theta)| &lt;= min_sidelobe   for theta outside the beam</span>
<span class="comment">%</span>
<span class="comment">% where y is the antenna array gain pattern (complex function) and</span>
<span class="comment">% variables are w (antenna array weights or shading coefficients).</span>
<span class="comment">% Gain pattern is a linear function of w: y(theta) = w'*a(theta)</span>
<span class="comment">% for some a(theta) describing antenna array configuration and specs.</span>
<span class="comment">%</span>
<span class="comment">% Written for CVX by Almir Mutapcic 02/02/06</span>

<span class="comment">% select array geometry</span>
ARRAY_GEOMETRY = <span class="string">'2D_RANDOM'</span>;
<span class="comment">% ARRAY_GEOMETRY = '1D_UNIFORM_LINE';</span>
<span class="comment">% ARRAY_GEOMETRY = '2D_UNIFORM_LATTICE';</span>

<span class="comment">%********************************************************************</span>
<span class="comment">% problem specs</span>
<span class="comment">%********************************************************************</span>
lambda = 1;           <span class="comment">% wavelength</span>
theta_tar = 60;       <span class="comment">% target direction</span>
half_beamwidth = 10;  <span class="comment">% half beamwidth around the target direction</span>
min_sidelobe = -20;   <span class="comment">% maximum sidelobe level in dB</span>

<span class="comment">%********************************************************************</span>
<span class="comment">% random array of n antenna elements</span>
<span class="comment">%********************************************************************</span>
<span class="keyword">if</span> strcmp( ARRAY_GEOMETRY, <span class="string">'2D_RANDOM'</span> )
  <span class="comment">% set random seed to repeat experiments</span>
  rand(<span class="string">'state'</span>,0);

  <span class="comment">% (uniformly distributed on [0,L]-by-[0,L] square)</span>
  n = 36;
  L = 5;
  loc = L*rand(n,2);

<span class="comment">%********************************************************************</span>
<span class="comment">% uniform 1D array with n elements with inter-element spacing d</span>
<span class="comment">%********************************************************************</span>
<span class="keyword">elseif</span> strcmp( ARRAY_GEOMETRY, <span class="string">'1D_UNIFORM_LINE'</span> )
  <span class="comment">% (unifrom array on a line)</span>
  n = 30;
  d = 0.45*lambda;
  loc = [d*[0:n-1]' zeros(n,1)];

<span class="comment">%********************************************************************</span>
<span class="comment">% uniform 2D array with m-by-m element with d spacing</span>
<span class="comment">%********************************************************************</span>
<span class="keyword">elseif</span> strcmp( ARRAY_GEOMETRY, <span class="string">'2D_UNIFORM_LATTICE'</span> )
  m = 6; n = m^2;
  d = 0.45*lambda;

  loc = zeros(n,2);
  <span class="keyword">for</span> x = 0:m-1
    <span class="keyword">for</span> y = 0:m-1
      loc(m*y+x+1,:) = [x y];
    <span class="keyword">end</span>
  <span class="keyword">end</span>
  loc = loc*d;

<span class="keyword">else</span>
  error(<span class="string">'Undefined array geometry'</span>)
<span class="keyword">end</span>

<span class="comment">%********************************************************************</span>
<span class="comment">% construct optimization data</span>
<span class="comment">%********************************************************************</span>
<span class="comment">% build matrix A that relates w and y(theta), ie, y = A*w</span>
theta = [1:360]';
A = kron(cos(pi*theta/180), loc(:,1)') + kron(sin(pi*theta/180), loc(:,2)');
A = exp(2*pi*i/lambda*A);

<span class="comment">% target constraint matrix</span>
[diff_closest, ind_closest] = min( abs(theta - theta_tar) );
Atar = A(ind_closest,:);

<span class="comment">% stopband constraint matrix</span>
ind = find(theta &lt;= (theta_tar-half_beamwidth) | <span class="keyword">...</span>
           theta &gt;= (theta_tar+half_beamwidth) );
As = A(ind,:);

<span class="comment">%********************************************************************</span>
<span class="comment">% optimization problem</span>
<span class="comment">%********************************************************************</span>
cvx_begin
  variable <span class="string">w(n)</span> <span class="string">complex</span>
  minimize( norm( w ) )
  subject <span class="string">to</span>
    Atar*w == 1;
    abs(As*w) &lt;= 10^(min_sidelobe/20);
cvx_end

<span class="comment">% check if problem was successfully solved</span>
disp([<span class="string">'Problem is '</span> cvx_status])
<span class="keyword">if</span> ~strfind(cvx_status,<span class="string">'Solved'</span>)
  <span class="keyword">return</span>
<span class="keyword">end</span>

fprintf(1,<span class="string">'The minimum norm of w is %3.2f.\n\n'</span>,norm(w));

<span class="comment">%********************************************************************</span>
<span class="comment">% plots</span>
<span class="comment">%********************************************************************</span>
figure(1), clf
plot(loc(:,1),loc(:,2),<span class="string">'o'</span>)
title(<span class="string">'Antenna locations'</span>)

<span class="comment">% plot array pattern</span>
y = A*w;

figure(2), clf
ymin = -30; ymax = 0;
plot([1:360], 20*log10(abs(y)), <span class="keyword">...</span>
     [theta_tar theta_tar],[ymin ymax],<span class="string">'r--'</span>,<span class="keyword">...</span>
     [theta_tar+half_beamwidth theta_tar+half_beamwidth],[ymin ymax],<span class="string">'g--'</span>,<span class="keyword">...</span>
     [theta_tar-half_beamwidth theta_tar-half_beamwidth],[ymin ymax],<span class="string">'g--'</span>,<span class="keyword">...</span>
     [0 theta_tar-half_beamwidth],[min_sidelobe min_sidelobe],<span class="string">'r--'</span>,<span class="keyword">...</span>
     [theta_tar+half_beamwidth 360],[min_sidelobe min_sidelobe],<span class="string">'r--'</span>);
xlabel(<span class="string">'look angle'</span>), ylabel(<span class="string">'mag y(theta) in dB'</span>);
axis([0 360 ymin ymax]);

<span class="comment">% polar plot</span>
figure(3), clf
zerodB = 50;
dBY = 20*log10(abs(y)) + zerodB;
plot(dBY.*cos(pi*theta/180), dBY.*sin(pi*theta/180), <span class="string">'-'</span>);
axis([-zerodB zerodB -zerodB zerodB]), axis(<span class="string">'off'</span>), axis(<span class="string">'square'</span>)
hold <span class="string">on</span>
plot(zerodB*cos(pi*theta/180),zerodB*sin(pi*theta/180),<span class="string">'k:'</span>) <span class="comment">% 0 dB</span>
plot( (min_sidelobe + zerodB)*cos(pi*theta/180), <span class="keyword">...</span>
      (min_sidelobe + zerodB)*sin(pi*theta/180),<span class="string">'k:'</span>)  <span class="comment">% min level</span>
text(-zerodB,0,<span class="string">'0 dB'</span>)
text(-(min_sidelobe + zerodB),0,sprintf(<span class="string">'%0.1f dB'</span>,min_sidelobe));
theta_1 = theta_tar+half_beamwidth;
theta_2 = theta_tar-half_beamwidth;
plot([0 55*cos(theta_tar*pi/180)], [0 55*sin(theta_tar*pi/180)], <span class="string">'k:'</span>)
plot([0 55*cos(theta_1*pi/180)], [0 55*sin(theta_1*pi/180)], <span class="string">'k:'</span>)
plot([0 55*cos(theta_2*pi/180)], [0 55*sin(theta_2*pi/180)], <span class="string">'k:'</span>)
hold <span class="string">off</span>
</pre>
<a id="output"></a>
<pre class="codeoutput">
 
Calling sedumi: 1439 variables, 414 equality constraints
   For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Put 2 free variables in a quadratic cone
eqs m = 414, order n = 1028, dim = 1441, blocks = 344
nnz(A) = 50003 + 0, nnz(ADA) = 54774, nnz(L) = 27594
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            2.69E+00 0.000
  1 :   4.26E-01 2.44E+00 0.000 0.9087 0.9000 0.9000  23.63  1  1  1.3E+00
  2 :   1.28E-01 2.25E+00 0.000 0.9217 0.9000 0.9000   8.54  1  1  9.2E-01
  3 :  -3.18E-01 1.61E+00 0.000 0.7162 0.9000 0.9000   7.27  1  1  2.9E-01
  4 :  -3.63E-01 9.78E-01 0.000 0.6066 0.9000 0.9000   2.60  1  1  1.7E-01
  5 :  -4.06E-01 5.71E-01 0.000 0.5843 0.9000 0.9000   1.79  1  1  1.4E-01
  6 :  -4.34E-01 3.87E-01 0.000 0.6778 0.9000 0.9000   1.30  1  1  1.0E-01
  7 :  -4.84E-01 2.20E-01 0.000 0.5674 0.9000 0.9000   1.13  1  1  5.6E-02
  8 :  -5.18E-01 1.48E-01 0.000 0.6722 0.9000 0.9000   0.95  1  1  3.8E-02
  9 :  -5.18E-01 2.20E-02 0.000 0.1490 0.9000 0.0000   0.97  1  1  1.6E-02
 10 :  -5.78E-01 2.80E-04 0.000 0.0127 0.9216 0.9000   0.93  1  1  3.2E-03
 11 :  -6.15E-01 3.82E-05 0.000 0.1366 0.8864 0.9000   0.85  1  1  1.4E-03
 12 :  -6.34E-01 1.70E-05 0.000 0.4441 0.9000 0.9000   0.94  1  1  6.1E-04
 13 :  -6.45E-01 5.58E-06 0.000 0.3287 0.9000 0.9000   0.98  1  1  2.0E-04
 14 :  -6.50E-01 1.59E-06 0.000 0.2857 0.9000 0.9000   0.99  1  1  5.8E-05
 15 :  -6.51E-01 3.76E-07 0.000 0.2358 0.9000 0.9000   1.00  1  1  1.4E-05
 16 :  -6.51E-01 1.01E-07 0.000 0.2692 0.9000 0.9000   1.00  1  1  3.7E-06
 17 :  -6.52E-01 2.16E-08 0.000 0.2131 0.9000 0.9000   1.00  1  1  7.9E-07
 18 :  -6.52E-01 4.32E-09 0.000 0.2007 0.9000 0.9000   1.00  2  2  1.6E-07
 19 :  -6.52E-01 8.25E-10 0.000 0.1908 0.9000 0.9000   1.00  2  2  3.0E-08
 20 :  -6.52E-01 8.00E-11 0.000 0.0970 0.9900 0.9900   1.00  2  2  2.9E-09

iter seconds digits       c*x               b*y
 20      0.3   Inf -6.5160741788e-01 -6.5160741756e-01
|Ax-b| =   2.9e-09, [Ay-c]_+ =   2.4E-09, |x|=  1.1e+01, |y|=  9.6e-01

Detailed timing (sec)
   Pre          IPM          Post
8.000E-02    3.300E-01    1.000E-02    
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 8.32687.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.651607
Problem is Solved
The minimum norm of w is 0.65.

</pre>
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<img src="ant_array_min_therm_noise__01.png" alt=""> <img src="ant_array_min_therm_noise__02.png" alt=""> <img src="ant_array_min_therm_noise__03.png" alt=""> 
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